A397558 - OEIS

A397558

Number of Hadamard-diagonalizable graphs of order 4*n.

4, 10, 4, 50, 4, 26, 4, 10196, 4

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OFFSET

1,1

COMMENTS

A (simple, undirected) graph G is called Hadamard-diagonalizable if there exists a Hadamard matrix H for which H^T*A*H is diagonal, where A is the adjacency matrix of G (A can equivalently be chosen to be the Laplacian matrix of G).

a(n) = 4 whenever n is odd and there exists a Hadamard matrix of order 4*n.

There is also one Hadamard-diagonalizable graph of order 1 (K_1) and two Hadamard-diagonalizable graphs of order 2 (K_2 and 2K_1).

Hadamard-diagonalizable graphs are necessarily regular (A005176) and integral (A077027).

LINKS

Table of n, a(n) for n=1..9.

J. Breen, S. Butler, M. Fuentes, B. Lidický, M. Phillips, A. W. N. Riasanovksy, S.-Y. Song, R. R. Villagrán, C. Wiseman, and X. Zhang, Hadamard diagonalizable graphs of order at most 36, The Electronic Journal of Combinatorics, 29, #P2.16 (2022). arXiv:2007.09235 [math.CO], 2020.

EXAMPLE

a(3) = 4 since there are 4 different 4*3 = 12-vertex Hadamard-diagonalizable graphs: K_{12}, K_{6,6}, 12K_1, and 2K_6.

CROSSREFS

Cf. A005176, A007299, A077027.

Sequence in context: A336988 A200454 A303052 * A003564 A347116 A389126

Adjacent sequences: A397555 A397556 A397557 * A397559 A397560 A397561

KEYWORD

nonn,hard,more

AUTHOR

Nathaniel Johnston, Jun 30 2026

STATUS

approved